Trees of visible components in the Mandelbrot set

Virpi Kauko

Fundamenta Mathematicae (2000)

  • Volume: 164, Issue: 1, page 41-60
  • ISSN: 0016-2736

Abstract

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We discuss the tree structures of the sublimbs of the Mandelbrot set M, using internal addresses of hyperbolic components. We find a counterexample to a conjecture by Eike Lau and Dierk Schleicher concerning topological equivalence between different trees of visible components, and give a new proof to a theorem of theirs concerning the periods of hyperbolic components in various trees.

How to cite

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Kauko, Virpi. "Trees of visible components in the Mandelbrot set." Fundamenta Mathematicae 164.1 (2000): 41-60. <http://eudml.org/doc/212447>.

@article{Kauko2000,
abstract = {We discuss the tree structures of the sublimbs of the Mandelbrot set M, using internal addresses of hyperbolic components. We find a counterexample to a conjecture by Eike Lau and Dierk Schleicher concerning topological equivalence between different trees of visible components, and give a new proof to a theorem of theirs concerning the periods of hyperbolic components in various trees.},
author = {Kauko, Virpi},
journal = {Fundamenta Mathematicae},
keywords = {combinatorial tree structure; Mandelbrot set; visible component},
language = {eng},
number = {1},
pages = {41-60},
title = {Trees of visible components in the Mandelbrot set},
url = {http://eudml.org/doc/212447},
volume = {164},
year = {2000},
}

TY - JOUR
AU - Kauko, Virpi
TI - Trees of visible components in the Mandelbrot set
JO - Fundamenta Mathematicae
PY - 2000
VL - 164
IS - 1
SP - 41
EP - 60
AB - We discuss the tree structures of the sublimbs of the Mandelbrot set M, using internal addresses of hyperbolic components. We find a counterexample to a conjecture by Eike Lau and Dierk Schleicher concerning topological equivalence between different trees of visible components, and give a new proof to a theorem of theirs concerning the periods of hyperbolic components in various trees.
LA - eng
KW - combinatorial tree structure; Mandelbrot set; visible component
UR - http://eudml.org/doc/212447
ER -

References

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  1. [At] P. Atela, Bifurcations of dynamic rays in complex polynomials of degree two, Ergodic Theory Dynam. Systems 12 (1991), 401-423. 
  2. [Be] A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems, Grad. Texts in Math. 132, Springer, 1991. 
  3. [BK] C. Bandt and K. Keller, Symbolic dynamics for angle-doubling on the circle II: Symbolic description of the abstract Mandelbrot set, Nonlinearity 6 (1993), 377-392. Zbl0785.58021
  4. [BS] H. Bruin and D. Schleicher, Symbolic Dynamics of Quadratic Polynomials, in preparation. 
  5. [CG] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext, Springer, 1993. Zbl0782.30022
  6. [K1] K. Keller, Correspondence and translation principles for the Mandelbrot set, preprint #14, Institute for Mathematical Sciences, Stony Brook, 1997. 
  7. [K2] K. Keller, Errata for Correspondence and translation principles for the Mandelbrot set', http://www.math-inf.uni-greifswald.de/~keller/research.html. 
  8. [LS] E. Lau and D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, preprint #19, Institute for Mathematical Sciences, Stony Brook, 1994, 
  9. [La] P. Lavaurs, Une description combinatoire de l'involution définie par M sur les rationnels à dénominateur impair, C. R. Acad. Sci. Paris 303 (1986), 143-146. Zbl0663.58018
  10. [Mi] J. Milnor, Periodic orbits, external rays, and the Mandelbrot set; an expository account, preprint #3, Institute for Mathematical Sciences, Stony Brook, 1999. 
  11. [Pe] C. Penrose, Quotients of the shift associated with dendrite Julia sets of quadratic polynomials, Ph.D. thesis, Warwick, 1990. 
  12. [S1] D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Ph.D. thesis, Cornell Univ., 1994. 
  13. [S2] D. Schleicher, Rational parameter rays of the Mandelbrot set, preprint #13, Institute for Mathematical Sciences, Stony Brook, 1997. 
  14. [Th] W. Thurston, On the geometry and dynamics of iterated rational maps, preprint, Princeton Univ., 1985. 

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